Armature is not modifying the mesh
In the first screenshot, it appears that your mesh has no vertex groups. As far as I am aware, skinning requires that the mesh has vertex groups with the same names as the bones in the armature, and cannot work otherwise. The vertices you want to be moved by a certain bone must be assigned to the matching vertex group. To add vertex groups to your mesh, click on the + button on the vertex groups panel (visible on the right side of your first screenshot). You can also automatically create the vertex groups by selecting the mesh, then shift selecting the armature, then hitting Ctrl + P and selecting Armature Deform with either Empty Groups, Envelope Weights, or Automatic Weights.